Stress-energy tensor

The stress-energy tensor T_{μν} is the symmetric (0,2) tensor field that encodes the local distribution and flow of energy and momentum in a relativistic field theory. Its components carry direct physical interpretations: T_{00} is the energy density, T_{0i} is the momentum density (equivalently, the energy flux), T_{ii} is the pressure (the diagonal spatial components), and T_{ij} for i ≠ j is the shear stress (the off-diagonal spatial components). Together these ten independent components describe how energy and momentum are distributed in space and how they flow across surfaces in spacetime. The conservation law ∇^μ T_{μν} = 0 is the relativistic generalisation of the conservation of energy and momentum — equivalent in flat spacetime to ∂^μ T_{μν} = 0, the condition that energy and three-momentum are conserved at every point.

The stress-energy tensor sits on the matter side of Einstein's field equations: G_{μν} = (8πG/c⁴) T_{μν}. Different matter sectors contribute different forms. A perfect fluid (no shear, isotropic pressure) has T_{μν} = (ρ + p/c²) u_μ u_ν − p g_{μν}, where ρ is the rest-frame energy density, p the pressure, and u^μ the fluid four-velocity. The electromagnetic field contributes T_{μν} = (1/μ₀)(F_{μλ} F_ν^λ − (1/4) g_{μν} F_{λρ} F^{λρ}) — quadratic in the field tensor F_{μν}. A scalar field, a Dirac spinor, dust, radiation, dark energy: each has a characteristic T_{μν}, and the sum of all sectors sources spacetime curvature via Einstein's equations. The conservation law is forced by the Bianchi-identity-implied divergence-freedom of the Einstein tensor — geometry demands that matter satisfy ∇^μ T_{μν} = 0.